Problem 33 - College Physics

Question

A wooden block of volume \(5.24 \times 10^{-4} \mathrm{~m}^{3}\) floats in water, and a small steel object of mass \(m\) is placed on top of the block. When \(m=0.310 \mathrm{~kg}\), the system is in equilibrium, and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by a second steel object with a mass less than \(0.310 \mathrm{~kg}\) ? What happens to the block when the steel object is replaced by yet another steel object with a mass greater than \(0.310 \mathrm{~kg}\) ?

Step-by-Step Solution

Answer

The density of the wooden block is dependent on the masses of the block and the water, as well as the volume of the block. The wooden block floats higher in the water when the steel object is replaced by a one with a smaller mass, and sinks deeper when it is replaced by a massive object.

Step 1: Finding the Density of the Wooden Block

To find the density of the wooden block, we need to apply the principle of buoyancy which states that the buoyant force on an object is equal to the weight of the water displaced by the object. In this case, since the system is in equilibrium, the buoyant force on the block plus the steel object equals the weight of the water displaced. We can use the formula \( \rho = \frac{m}{v} \) where \( \rho \) is the density, \( m \) is the mass and \( v \) is the volume. The mass of the water displaced is equivalent to the sum of the mass of wood and steel, which we can write as \( m_{wood} = m_{water} - m_{steel} \). Substituting these into the formula gives us the density of the wooden block.

Step 2: Effect of Replacing the Steel Object

When the steel object is replaced by a second one with a smaller mass, the total weight of the system decreases. This means that less water needs to be displaced to provide the buoyant force, and as a result, the block floats higher in the water than before. On the other hand, if the steel object is replaced by one with a greater mass, the total weight of the system increases and thus more water needs to be displaced for the buoyant force. As a result, the block will sink deeper in the water.

Key Concepts

Buoyant Force

The buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is what allows ships, for instance, to float on the surface of the water instead of sinking. If we refer to our textbook exercise, the wooden block floats on water because the buoyant force acts upward, balancing the downward force of gravity on the block and the added steel object.

According to the principle of buoyancy, this force is equal to the weight of the fluid that is displaced by the object. The very essence of how and why things float or sink can be attributed to this force. When a body is partially or completely submerged, the fluid exerts pressure over the surface of the body. This pressure difference results in a net upward force. If this buoyant force is greater than the object's weight, it will float; otherwise, it will sink.

Equilibrium in Fluids

When we talk about equilibrium in fluids, we're focusing on a situation where the net force on an object in a fluid is zero. In the context of the exercise, the equilibrium is achieved when the wooden block floats without sinking further or rising, which happens when the buoyant force and gravity are balanced.

For an object to be in equilibrium in a fluid, the object's weight must be exactly balanced by the buoyant force. If these forces are equal and opposite, the object won't move and remains at rest within the fluid. This balance is what allows boats to remain afloat at a constant level and is crucial for many real-life applications, including the design of ships and underwater vessels.

Principle of Buoyancy

The principle of buoyancy underlines the conditions under which an object will float, sink, or remain suspended in a fluid. It indicates that the buoyant force experienced by an object in a fluid is equal to the weight of fluid it displaces. This principle is fundamental in understanding how different objects interact with fluids, be they gases or liquids.

To exemplify this principle, consider our wooden block exercise. The block floats with the small steel object on top because the mass of the water displaced is equal to the total mass of the block and steel object combined. If the steel object changes, the displacement and therefore the buoyant force would change accordingly, altering the equilibrium state of our system.

Archimedes' Principle

Finding the Density with Archimedes' Principle

Archimedes' principle is a fundamental law of physics that states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces. This principle applies to all fluids and is essential for solving a variety of problems involving buoyancy.

In relation to our wooden block problem from the textbook, Archimedes' principle is used to calculate the density of the wood. The block, in conjunction with the steel object's mass, displaces a certain amount of water equal to their combined weight when in equilibrium. Hence, this principle provides the basis on which we computed the density of the wooden block, using the formula for density \( \rho = \frac{m}{v} \).